# math identity?

Determine whether 4sin (x) cos (x) +tan^2 (x) +1 = 1/(cos^2(x)) + sin(4x) is an identity

### 3 Answers

- Pramod KumarLv 74 months ago
if this equation/identity is not satisfied by even one value of x , it is an equation -

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Let x = 0

4sin (x) cos (x) +tan^2 (x) +1 = 1/(cos^2(x)) + sin(4x)

=> 4 sin (0) cos (0) + tan²(0) + 1 = 1/cos²0 + sin (4*0)

=> 0 + 0 + 1 = 1 + 0 ................ True

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Let x = 30°

4sin (x) cos (x) +tan^2 (x) +1 = 1/(cos^2(x)) + sin(4x)

=> 4 sin 30 * cos 30 + tan²(30) + 1 = 1/cos²30 + sin (4*30)

=>√3 + 1/3 + 1 ≠ 4/3 + √3/2 ....................

Hence this is an equation.

- L. E. GantLv 74 months ago
It isn't!

1/cos^2(x) = sec^2(x) = tan^2(x) + 1 ... one of the Pythagorean identities4sin(x)cos(x) = 2* 2sin(x)cos(x) = 2sin(2x)... sin(2x) = 2sin(x)cos(x) is a standard identity

so sec^2(x) + 2sin(2x) = 4sin(x)cos(x) + tan^2(x) + 1

would be the identity.

- cosmoLv 74 months ago
Not an identity. It's only true for x = N pi/2, it's not true for any other value of x. So, consider x = pi/4 it's a counter-example, the left side = 4 and the right side = 2.

It's true to first order at N pi/2.